Given the integral, ∫ ( x ^ 3 +1)(x 2) x ^ 2 #160; dx =∫ x ^ 4 +x 2 x ^ 3 2 x ^ 2 #160; dx =∫[ x ^ 2 + 1 x 2x 2 x ^ 2 #160; ] ...

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Integration by Substitution

Solve: ∫x3+1...

Question

Solve: $\int \frac{(x^{3} + 1) (x - 2)}{x^{2}} . d x$

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Solution

Given the integral,
$\int \frac{(x^{3} + 1) (x - 2)}{x^{2}} d x = \int \frac{x^{4} + x - 2 x^{3} - 2}{x^{2}} d x = \int [x^{2} + \frac{1}{x} - 2 x - \frac{2}{x^{2}}] d x = \int x^{2} d x + \int \frac{1}{x} d x - 2 \int x d x - 2 \int \frac{1}{x^{2}} d x = \frac{x^{3}}{3} + ln (| x |) - \frac{2 x^{2}}{2} - 2 (- \frac{1}{x}) + C = ln (| x |) + \frac{x^{3}}{3} - x^{2} + \frac{2}{x} + C = ln (| x |) + \frac{x^{3} - 3 x^{2}}{3} + \frac{2}{x} + C .$

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